Obtaining and using second order derivative information in the solution of large scale inverse problems

• Authors: Mihai Alexe;Alexandru Cioaca;Adrian Sandu
• Venue: SpringSim '10 Proceedings of the 2010 Spring Simulation Multiconference
• Year: 2010

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost function subject to the model constraints. The numerical solution of such optimization problems requires the derivatives of a chosen cost function I dependent on the model parameters. Given that the parameter space is large in real-life problems, the derivatives of I can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of products between the Hessian of the cost functional and a user defined vector. In this paper we review the mathematical foundations of the second order adjoint sensitivity method. We then evaluate their performance in several data assimilation, sensitivity analysis, and uncertainty quantification scenarios, for a two dimensional shallow water flow simulation. In the data assimilation problem, we compare the performance of several well-known optimization methods that make use of first and second order information.